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Theorem ntrclscls00 41206
Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that the closure of the empty set is the empty set hold equally. (Contributed by RP, 1-Jun-2021.)
Hypotheses
Ref Expression
ntrcls.o 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
ntrcls.d 𝐷 = (𝑂𝐵)
ntrcls.r (𝜑𝐼𝐷𝐾)
Assertion
Ref Expression
ntrclscls00 (𝜑 → ((𝐼𝐵) = 𝐵 ↔ (𝐾‘∅) = ∅))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘   𝑗,𝐼,𝑘   𝑗,𝐾,𝑘   𝜑,𝑖,𝑗,𝑘
Allowed substitution hints:   𝐷(𝑖,𝑗,𝑘)   𝐼(𝑖)   𝐾(𝑖)   𝑂(𝑖,𝑗,𝑘)

Proof of Theorem ntrclscls00
StepHypRef Expression
1 ntrcls.o . . . . . 6 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
2 ntrcls.d . . . . . 6 𝐷 = (𝑂𝐵)
3 ntrcls.r . . . . . 6 (𝜑𝐼𝐷𝐾)
41, 2, 3ntrclsfv1 41195 . . . . 5 (𝜑 → (𝐷𝐼) = 𝐾)
54fveq1d 6670 . . . 4 (𝜑 → ((𝐷𝐼)‘∅) = (𝐾‘∅))
62, 3ntrclsbex 41174 . . . . 5 (𝜑𝐵 ∈ V)
71, 2, 3ntrclsiex 41193 . . . . 5 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
8 eqid 2738 . . . . 5 (𝐷𝐼) = (𝐷𝐼)
9 0elpw 5219 . . . . . 6 ∅ ∈ 𝒫 𝐵
109a1i 11 . . . . 5 (𝜑 → ∅ ∈ 𝒫 𝐵)
11 eqid 2738 . . . . 5 ((𝐷𝐼)‘∅) = ((𝐷𝐼)‘∅)
121, 2, 6, 7, 8, 10, 11dssmapfv3d 41157 . . . 4 (𝜑 → ((𝐷𝐼)‘∅) = (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))))
135, 12eqtr3d 2775 . . 3 (𝜑 → (𝐾‘∅) = (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))))
14 dif0 4259 . . . . . . 7 (𝐵 ∖ ∅) = 𝐵
1514fveq2i 6671 . . . . . 6 (𝐼‘(𝐵 ∖ ∅)) = (𝐼𝐵)
16 id 22 . . . . . 6 ((𝐼𝐵) = 𝐵 → (𝐼𝐵) = 𝐵)
1715, 16syl5eq 2785 . . . . 5 ((𝐼𝐵) = 𝐵 → (𝐼‘(𝐵 ∖ ∅)) = 𝐵)
1817difeq2d 4011 . . . 4 ((𝐼𝐵) = 𝐵 → (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))) = (𝐵𝐵))
19 difid 4257 . . . 4 (𝐵𝐵) = ∅
2018, 19eqtrdi 2789 . . 3 ((𝐼𝐵) = 𝐵 → (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))) = ∅)
2113, 20sylan9eq 2793 . 2 ((𝜑 ∧ (𝐼𝐵) = 𝐵) → (𝐾‘∅) = ∅)
22 pwidg 4507 . . . . 5 (𝐵 ∈ V → 𝐵 ∈ 𝒫 𝐵)
236, 22syl 17 . . . 4 (𝜑𝐵 ∈ 𝒫 𝐵)
241, 2, 3, 23ntrclsfv 41199 . . 3 (𝜑 → (𝐼𝐵) = (𝐵 ∖ (𝐾‘(𝐵𝐵))))
2519fveq2i 6671 . . . . . 6 (𝐾‘(𝐵𝐵)) = (𝐾‘∅)
26 id 22 . . . . . 6 ((𝐾‘∅) = ∅ → (𝐾‘∅) = ∅)
2725, 26syl5eq 2785 . . . . 5 ((𝐾‘∅) = ∅ → (𝐾‘(𝐵𝐵)) = ∅)
2827difeq2d 4011 . . . 4 ((𝐾‘∅) = ∅ → (𝐵 ∖ (𝐾‘(𝐵𝐵))) = (𝐵 ∖ ∅))
2928, 14eqtrdi 2789 . . 3 ((𝐾‘∅) = ∅ → (𝐵 ∖ (𝐾‘(𝐵𝐵))) = 𝐵)
3024, 29sylan9eq 2793 . 2 ((𝜑 ∧ (𝐾‘∅) = ∅) → (𝐼𝐵) = 𝐵)
3121, 30impbida 801 1 (𝜑 → ((𝐼𝐵) = 𝐵 ↔ (𝐾‘∅) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1542  wcel 2113  Vcvv 3397  cdif 3838  c0 4209  𝒫 cpw 4485   class class class wbr 5027  cmpt 5107  cfv 6333  (class class class)co 7164  m cmap 8430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-ov 7167  df-oprab 7168  df-mpo 7169  df-1st 7707  df-2nd 7708  df-map 8432
This theorem is referenced by: (None)
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